# Book:Haïm Brezis/Functional Analysis, Sobolev Spaces and Partial Differential Equations

## Haïm Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations

Published $2010$, Springer

ISBN 978-0387709130.

### Contents

Preface

1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions
1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functionals
1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets
1.3 The Bidual $E^{**}$. Orthogonality Relations
1.4 A Quick Introduction to the Theory of Conjugate Convex Functions
Exercises for Chapter 1
2. The Uniform Boundedness Principle and the Closed Graph Theorem
2.1 The Baire Category Theorem
2.2 The Uniform Boundedness Principle
2.3 The Open Mapping Theorem and the Closed Graph Theorem
2.4 Complimentary Subspaces. Right and Left Invertability of Linear Operators
2.5 Orthogonality revisited
2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint
2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators
Exercises for Chapter 2
3. Weak Topologies, Reflexive Spaces, Separable Spaces, Uniform Convexity
3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous
3.2 Definition and Elementary Properties of the Weak Topology $\map \sigma {E,E^*}$
3.3 Weak Topology, Convex Sets and Linear Operators
3.4 The Weak* Topology $\map \sigma {E^*,E}$
3.5 Reflexive Spaces
3.6 Separable Spaces
3.7 Uniformly Convex Spaces
Exercises for Chapter 3
4. $L^p$ Spaces
4.1 Some Results about Integration That Everyone Must Know
4.2 Definition and Elemenary Properties of $L^p$ Spaces
4.3 Reflexivity. Separability. Dual of $L^p$
4.4 Convolution and regularization
4.5 Criterion for Strong Compactness in $L^p$
Exercises for Chapter 4
5. Hilbert Spaces
5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set
5.2 The Dual Space of a Hilbert Space
5.3 The Theorems of Stampacchia and Lax-Milgram
5.4 Hilbert sums. Orthonormal Bases
Exercises for Chapter 5
6. Compact Operators, Spectral Decomposition of Self-Adjoint Compact Operators
6.2 The Riesz-Fredholm Theory
6.3 The Spectrum of a Compact Operator
6.4 Spectral Decomposition of Self-Adjoint Compact Operators
Exercises for Chapter 6
7. The Hille-Yosida Problem
7.1 Definition and Elementary Properties of Maximal Monotone Operators
7.2 Solution of the Evolution Problem $\dfrac {\d u} {\d t} + A u = 0$ on $\hointr 0 {+ \infty}$, $\map u 0 = u_0$. Existence and uniqueness
7.3 Regularity
8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension
8.1 Motivation
8.2 The Sobolev Space $\map {W^{1,p} } {I}$
8.3 The Space $W_0^{1,p}$
8.4 Some Examples of Boundary Value Problems
8.5 The Maximum Principle
8.6 Eigenfunctions and Spectral Decomposition
Exercises for Chapter 8
9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in $N$ Dimensions
9.1 Definition and Elementary Properties of the Sobolev Spaces $\map {W^{1,p}} {\Omega}$
9.2 Extension Operators
9.3 Sobolev Inequalities
9.4 The Space $\map {W_0^{1,p} } \Omega$
9.5 Variational Formulation of Some Boundary Value Problems
9.6 Regularity of Weak Solutions
9.7 The Maximum Principle
9.8 Eigenfunctions and Spectral Decomposition
10. Evolution Problems: the Heat Equation and the Wave Equation
10.1 The Heat Equation: Existence, Uniqueness, and Regularity
10.2 The Maximum Principle
10.3 The Wave Equation
11. Miscellaneous Complements
11.1 Finite-Dimensional and Finite-Codimensional Spaces
11.2 Quotien Spaces
11.3 Some Classical Spaces and Sequences
11.4 Banach Space over $\C$: What Is Similar and What Is Different

Solutions of Some Exercises

Problems

Partial Solutions

Notation

References

Index